Find All Complex Solutions 0=X^2+7X+12

$0 = X^{2} + 7 X + 12$

Move all the expressions to the left side of the equation.

Subtract $X^{2}$ from both sides of the equation.

$- X^{2} = 7 X + 12$

Subtract $7 X$ from both sides of the equation.

$- X^{2} - 7 X = 12$

Subtract $12$ from both sides of the equation.

$- X^{2} - 7 X - 12 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = - 1$ , $b = - 7$ , and $c = - 12$ into the quadratic formula and solve for $X$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (- 1 \cdot - 12)}}{2 \cdot - 1}$

Simplify.

Simplify the numerator.

Raise $- 7$ to the power of $2$ .

$X = \frac{7 \pm \sqrt{49 - 4 \cdot - 1 \cdot - 12}}{2 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$X = \frac{7 \pm \sqrt{49 + 4 \cdot - 12}}{2 \cdot - 1}$

Multiply $4$ by $- 12$ .

$X = \frac{7 \pm \sqrt{49 - 48}}{2 \cdot - 1}$

Subtract $48$ from $49$ .

$X = \frac{7 \pm \sqrt{1}}{2 \cdot - 1}$

Any root of $1$ is $1$ .

$X = \frac{7 \pm 1}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$X = \frac{7 \pm 1}{- 2}$

Move the negative in front of the fraction.

$X = - \frac{7 \pm 1}{2}$

The final answer is the combination of both solutions.

$X = - 4, - 3$